Magnetic Resonance Imaging (MRI) is a powerful biomedical tool to non-invasively image the human body for disease diagnosis and healthy physiological process research. MRI is intrinsically slow, and numerous methods have been proposed to accelerate the MRI scan. One of the most important acceleration method is the under-sampling reconstruction technique, where fewer samples are acquired in the MRI data space (k-space), and prior knowledge is used to restore the images. The MRI image reconstruction problem is often formulated as an optimization problem with constraints, and iterative algorithms, such as non-linear conjugate gradient (NLCG), fast iterated shrinkage/thresholding algorithm (FISTA), alternating direction method of multipliers (ADMM), Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method, or the like. are used to solve the optimization problem.